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JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced PDF Download

2023

Q1: Let the position vectors of the points P, Q, R and S be JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced respectively. Then which of the following statements is true?
(a) The points P, Q, R and S are NOT coplanar
(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced is the position vector of a point which divides PR internally in the ratio 5 : 4
(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced is the position vector of a point which divides PR externally in the ratio 5 : 4
(d) The square of the magnitude of the vector JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced is 95    [JEE Advanced 2023 Paper 2]
Ans:
(b) 

Q2: Let ℓ1 and ℓ2 be the lines JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced respectively. Let X be the set of all the planes H that contain the line ℓ1. For a plane H, let d(H) denote the smallest possible distance between the points of ℓ2 and H. Let H0 be a plane in X for which d(H0) is the maximum value of d(H) as H varies over all planes in X.
Match each entry in List-I to the correct entries in List-II. 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

The correct option is:
(a) (P)→(2)(Q)→(4)(R)→(5)(S)→(1)
(b) (P)→(5)(Q)→(4)(R)→(3)(S)→(1)
(c) (P)→(2)(Q)→(1)(R)→(3)(S)→(2)
(d) (P)→(5)(Q)→(1)(R)→(4)(S)→(2)                           [JEE Advanced 2023 Paper 1]
Ans:
(b)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

For plane
d(H)= Smallest possible distance between the points of 2 and Plane.
d(H0) = Maximum value of d(H)
For   d(H0)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced


2 is Parallel to plane containing 1
Equation of plane

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

 It contain 1
∴ a + b + c = 0 ..........(1)
For largest possible distance between plane (1) and 2 the line 2 must be parallel to plane (1)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

∴ Point of intersection (1, 1, 1) Distance from origin
= JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Q3: Let P be the plane JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and let JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and the distance of (α, β, γ) from the plane P is 7/2. Let JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced be three distinct vectors in S such that JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced. Let S be the volume of the parallelepiped determined by vectors JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced. Then the value of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced is :                   [JEE Advanced 2023 Paper 1]
Ans: 
45

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are elements of set S and in set S magnitude of vector is 1
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are unit vectors and by equation (1) we can system JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are equally inclined and vertices of equilateral triangle also lying on a circle which is intersection of sphere 
Distance from Origin to P

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Equation of the plane is

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Equation of sphere = JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

∴ Radius or circleJEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

∴ Area or triangle = JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Velocity of Parallelepiped,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

= 45

 

2022


Q1: Let JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced be the unit vectors along the three positive coordinate axes. Let

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

be three vectors such that JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Then, which of the following is/are TRUE?
(a) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(d) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced                      [JEE Advanced 2022 Paper 2]
Ans:
(b), (c) & (d)
Given,
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Adding (1), (2) and (3), we get

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Aso given,
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now,
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

And

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Comparing value of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced with equation (4), we get 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Multiplying both side with JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced, we get

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

∴ B is correct

As, JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Option (D) is correct. 

Given, JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

∴ Option (C) is correct.

Q2: Let P1 and P2 be two planes given by

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1 and P2 ? 
(a) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

(d) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced                             [JEE Advanced 2022 Paper 1]

Ans: (a), (b) & (d)
P1 and P2 be two planes given by 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now finding line of intersection of both the planes,
Let z = λ, then 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now solving the eq. (1) and (2) we get, 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now any skew line with the line of intersection of given plane can be edge of tertrahedron.
Now using above concept we will solve all options.

For option (A)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now satisfying this point in given plane we have,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now we can see line is intersecting the plane P1, at some point.
Now checking for plane (P2)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Also intersecting plane (P2)
Hence, it can be the edge of tetrahedron.

For option (B)

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

this point is satisfying plane P1

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now checking for plane P2

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, it can be the edge of tetrahedron. 

For option (D),

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

point (λ, −2λ + 4, 3λ) and for λ = 0 point will be (0, −4, 0) which is lying on line of intersection and DR of plane P2 is (−2,5,4) and DR of line is (1,−2,3)
Now line is lying completely on P2
Hence, it can be the edge of tetrahedron. 

Q3: Let S be the reflection of a point Q with respect to the plane given by

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

where t, p are real parameters and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced respectively, then which of the following is/are TRUE ?
(a) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(d)
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced            [JEE Advanced 2022 Paper 1]
Ans: (a), (b) & (c)
Given : equation of plane

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

on rearranging we get,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

So, equation of plane in standard form is given by

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now given co-ordinate of Q = (10, 15, 20)
And Co-ordinates of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now using the image formula of point and plane we get,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now solving all options.

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

2021

Q1: Let αβ and γ be real numbers such that the system of linear equations
x + 2y + 3z = α
4x + 5y + 6z = β
7x + 8y + 9z = γ  1
is consistent. Let | M | represent the determinant of the matrix

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
Let P be the plane containing all those (αβ, γ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
The value of | M | is _________.                  [JEE Advanced 2021]
Ans: 
1
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

On equating the coefficients,
4A + B = 7 .... (i)
5A + 2B = 8 .... (ii)
and  (γ  1) =  Aβ  αB ..... (iii) 
On solving Eqs. (i) and (ii), we get A = 2 and B = 1
From Eq. (iii), we get

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, determinant of

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced [from Eq. (iv)]
Q2: Let α, β and γ be real numbers such that the system of linear equations
x + 2y + 3z = α
4x + 5y + 6z = β
7x + 8y + 9z = γ − 1
is consistent. Let | M | represent the determinant of the matrix

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
Let P be the plane containing all those (α, β, γ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
The value of | D | is _________.                  [JEE Advanced 2021]
Ans: 
1.5
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

On equating the coefficients,
4A + B = 7 .... (i)
5A + 2B = 8 .... (ii)
and − (γ − 1) = − Aβ − αB ..... (iii)
On solving Eqs. (i) and (ii), we get A = 2 and B = −1
From Eq. (iii), we get

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, determinant of

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced [from Eq. (iv)]

Equation of plane P is given by x −2y + z = 1
Hence, perpendicular distance of the point (0, 1, 0) from the plane 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

2020

Q1: Let α2 + β2 + γ≠ 0 and α + γ = 1. Suppose the point (3, 2, −1) is the mirror image of the point (1, 0, −1) with respect to the plane αx + βy + γz = δ. Then which of the following statements is/are TRUE?
(a) α + β = 2
(b) δ − γ = 3
(c) δ + β = 4
(d) α + β + γ = δ`          [JEE Advanced 2020 Paper 2]
Ans:
(a), (b) & (c)
Since, the point A(3, 2, 1) is the mirror image of the point B(1, 0, 1) with respect to the plane αx + βy + γz = δ, then
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

it is given that JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

And, the mid-point of AB, M(2, 1, −1) lies on the given plane, so

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Q2: Let a and b be positive real numbers. Suppose JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are adjacent sides of a parallelogram PQRS. Let u and v be the projection vectors of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced along PQ and PS, respectively. If |u| + |v| = |w| and if the area of the parallelogram PQRS is 8, then which of the following statements is/are TRUE?
(a) a + b = 4
(b) a − b = 2
(c) The length of the diagonal PR of the parallelogram PQRS is 4
(d) w is an angle bisector of the vectors PQ and PS              [JEE Advanced 2020 Paper 2]
Ans: 
(a) & (c)
Given vectors  JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced are adjacent sides of a parallelogram PQRS. 

so area of parallelogram PQRS =
|PQ × PS| = 2ab = 8 (given)
 ab = 4 ......(i)
According to the question,
|u| = |projection vector of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced along PQ| 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

and, similarly, |v| = |projection vector of JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced along PS| 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

From Eqs. (i) and (ii), we get

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

and the length of diagonal PR

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

And, the angle bisector of vector PQ and PS is along the vector

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

2019

Q1: Three lines JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear? 
(a) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(d) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced             [JEE Advanced 2019 Paper 2]
Ans:
(c) & (d)
Given lines,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, let the point P on L1 = (λ, 0, 0)
the point Q on L2 = (0, μ, 1), and
the point R on L3 = (1, 1, v)
For collinearity of points P, Q and R, there should be a non-zero scalar 'm', such that PQ = m PR 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, Q can not have coordinator (0, 0, 1 ) and (0, 1, 1)
Hence, options (c) and (d) are correct.

Q2: Let L1 and L2 denote the lines 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3? 
(a) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(b) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(c) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced
(d) JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced              [JEE Advanced 2019 Paper 1]
Ans:
(a), (b) & (c)
Given lines,
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

and since line L3 is perpendicular to both lines L1 and L2.
Then a vector along L3 will be, 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, let a general point on line L1.
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and on line L2 as JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced and let P and Q are point of intersection of lines L1, L3 and L2, L3, so direction ratio's of L3 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

[from Eqs. (i) and (ii)]
JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, we can take equation of line L3 as JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced, where a is position vector of any point on line L3 and possible vector of a are 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, options (a), (b) and (c) are correct.

2018

Q1: Let P1 : 2x + y  z = 3 and P2 : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE? 
(a) The line of intersection of P1 and P2 has direction ratios 1, 2, −1

(b) The line JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advancedis perpendicular to the line of intersection of P1 and P2
(c) The acute angle between P1 and P2 is 60∘
(d) If P3 is the plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of P1 and P2, then the distance of the point (2, 1, 1) from the plane P3 is 2/√3     [JEE Advanced 2018 Paper 1]
Ans: (
c) & (d)
We have,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Here, JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

and JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

(a) Direction ratio of the line of intersection of P1 and P2 is JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, statement a is false.
(b) We have,

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

his line is parallel to the line of intersection of P1 and P2.
Hence, statement (b) is false.
(c) Let acute angle between P1 and P2 be θ.
We know that, 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, statement (c) is true.

(d) Equation of plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of P1 and P2 is

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Now, distance of the point (2, 1, 1) from the plane x  y + z = 0 is 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Hence, statement (d) is true.

Q2: Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is _____.                    [JEE Advanced 2018 Paper 2]
Ans:
8
Let P(α, β, γ) and R is image of P in the XY-plane.
 R(α, β, -γ)
Also, Q is the image of P in the plane x + y = 3 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Since, Q is lies on Z-axis 

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Given, distance of P from X-axis be 5

JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

Then, JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced

The document JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on JEE Advanced Previous Year Questions (2018 - 2023): Vector Algebra and 3D Geometry - Mathematics (Maths) for JEE Main & Advanced

1. What are the key topics covered in the JEE Advanced Vector Algebra and 3D Geometry section?
Ans. The key topics covered in the JEE Advanced Vector Algebra and 3D Geometry section include vectors, vector algebra, 3D geometry, straight lines, planes, and coordinate geometry.
2. How important is the Vector Algebra and 3D Geometry section in the JEE Advanced exam?
Ans. The Vector Algebra and 3D Geometry section is considered important in the JEE Advanced exam as it carries significant weightage and can help students score well if they have a good understanding of the concepts.
3. What are some common types of questions asked in the Vector Algebra and 3D Geometry section of JEE Advanced?
Ans. Common types of questions asked in this section include finding the equation of a line or plane, calculating the angle between two vectors, determining the distance between two points in space, and solving problems involving intersections of lines and planes.
4. How can students effectively prepare for the Vector Algebra and 3D Geometry section of JEE Advanced?
Ans. Students can effectively prepare for this section by thoroughly understanding the basic concepts, practicing a variety of problems, solving previous year question papers, and seeking help from teachers or mentors when needed.
5. Are there any specific tips or tricks to tackle challenging questions in the Vector Algebra and 3D Geometry section of JEE Advanced?
Ans. Some tips to tackle challenging questions in this section include breaking down the problem into smaller parts, visualizing the problem using diagrams, practicing regularly, and revisiting fundamental concepts to strengthen problem-solving skills.
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